Factor completely. [tex]$b^2-12 b+36$[/tex]
Answer (2)
The quadratic expression b 2 − 12 b + 36 factors to ( b − 6 ) 2 . This is done by recognizing it as a perfect square trinomial and verifying through expansion. The complete factorization is thus ( b − 6 ) ( b − 6 ) .
;
Recognize the quadratic expression as a perfect square trinomial.
Identify the values that fit the perfect square trinomial pattern.
Rewrite the expression in factored form.
Verify the factorization by expanding the factored form to ensure it matches the original expression. The factored form is ( b − 6 ) 2 .
Explanation
Understanding the Problem We are given the quadratic expression b 2 − 12 b + 36 and asked to factor it completely.
Recognizing the Pattern We recognize that the given expression is a perfect square trinomial of the form a 2 − 2 ab + b 2 = ( a − b ) 2 .
Identifying 'a' and 'b' In our expression, a = b . We need to find a value for b such that 2 ⋅ b ⋅ b = 12 b and b 2 = 36 .
Determining the Value of b Since b 2 = 36 , we have b = 6 .
Rewriting the Expression Therefore, we can rewrite the expression as ( b − 6 ) 2 .
Verifying the Factorization To verify our factorization, we expand ( b − 6 ) 2 : ( b − 6 ) 2 = ( b − 6 ) ( b − 6 ) = b 2 − 6 b − 6 b + 36 = b 2 − 12 b + 36 This matches the original expression.
Final Factorization The complete factorization is ( b − 6 ) ( b − 6 ) or ( b − 6 ) 2 .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to design structures and predict their behavior under different loads. Architects use factoring to create aesthetically pleasing and structurally sound buildings. Financial analysts use factoring to model investment portfolios and assess risk. By mastering factoring, you'll be equipped to solve a wide range of problems in various fields.
